
\chapter{格林函数法}

\section{习题1-5}

\begin{yyEx}
	求解上半平面的狄利克雷问题
	\begin{equation*}
		\begin{dcases}
			u_{xx}+u_{yy} = 0,&y>0,\\
			u|_{y=0} = f(x).&~
		\end{dcases}
	\end{equation*}
\end{yyEx}

\begin{yyEx}
	求解上半空间的狄利克雷问题
	\begin{equation*}
	\begin{dcases}
	\nabla^2u = 0,&z>0,\\
	u|_{z=0} = f(x,y).&~
	\end{dcases}
	\end{equation*}
\end{yyEx}

\begin{yyEx}
	\begin{enumerate}
		\item 用电像法求出圆域泊松方程的格林函数$G(x,y;x_0,y_0)$, $M_0(x_0,y_0)$是圆内的一点, $G$满足
		\begin{equation*}
			\nabla^2G = \delta(x-x_0)\delta(y-y_0),~~G|_{\rho = a} = 0.
		\end{equation*}
		\item 在圆形域$\rho\leqslant a$上求拉普拉斯方程第一边值问题:
		\begin{equation*}
		\begin{dcases}
			\nabla^2u = 0,&\rho\leqslant a,\\
			u|_{\rho = a} = f(\varphi).&~
		\end{dcases}
		\end{equation*}
		\item 在圆形域$\rho\leqslant a$上求解$\nabla^2u = 0$,使满足边界条件$u|_{\rho = a} = A\cos\varphi$.
	\end{enumerate}
\end{yyEx}

\begin{yyEx}
	求区域$0\leqslant x<\infty,0\leqslant y<\infty$上的格林函数,并由此求解狄利克雷问题:
	\begin{equation*}
	\begin{dcases}
	u_{xx}+u_{yy} = 0,&~\\
	u(0,y) = f(y),&0\leqslant y\leqslant\infty\\
	u(x,0) = 0,&0\leqslant y\leqslant\infty,
	\end{dcases}
	\end{equation*}
	其中$f$为已知的连续函数,且$f(0) = 0$.
\end{yyEx}

\begin{yyEx}
	求矩形区域内泊松方程的狄利克雷边值问题的格林函数
	\begin{equation*}
	\begin{dcases}
	&\nabla^2G = -\delta(x-x_0)\delta(y-y_0),~\\
	&G|_{x= 0} = 0,~~G|_{x = a} = 0,\\
	&G|_{y= 0} = 0,~~G|_{x = b} = 0.
	\end{dcases}
	\end{equation*}
\end{yyEx}